Convex polygons. Definition nke a convex polygon. The diagonals nke a convex polygon

Ndị a geometric shapes niile gbara anyị gburugburu. Convex polygons bụ eke, dị ka a n'ụgbụgbọ mmanụ aṅụ ma ọ bụ aka wuru (nwoke mere). A na ọnụ ọgụgụ na-eji na-amị dị iche iche nke coatings na nkà, ihe owuwu, ịchọ mma, wdg Convex polygons nwere ihe onwunwe na ha ihe na-edina na otu akụkụ nke a ogologo akara na-aga site na ụzọ nke n'akụkụ vertices nke geometrical ọgụgụ. E nwere ndị ọzọ nkọwa. Ọ na-akpọ convex polygon, nke na-mere ndokwa na a otu ọkara ụgbọelu na-akwanyere ihe ọ bụla ogologo akara nwere otu onye nke n'akụkụ.

convex polygons

Na N'ezie nke elementrị jiometrị na-mgbe na-emeso nnọọ mfe polygons. Iji ghọta ihe Njirimara nke geometric shapes gị mkpa ịghọta ha ọdịdị. Iji malite ịghọta na mechiri emechi bụ ihe ọ bụla akara onye nsọtụ bụ otu. Na ọnụ ọgụgụ kpụrụ ya, nwere ike a dịgasị iche iche nke nhazi. Polygon a na-akpọ dị mfe mechiri emechi polyline onye n'akụkụ nkeji na-adịghị dị na n'otu ahịrị kwụ ọtọ. Ya na njikọ na na ọnụ na-, karị, n'akụkụ na n'elu geometrical ọgụgụ. A mfe polyline ga bụghị irutu onwe ya.

vertices nke polygon na-akpọ ndị agbata obi, bụrụ na ha na nsọtụ nke onye nke ya n'akụkụ. A geometric ọgụgụ, nke nwere a n-nke ọnụ ọgụgụ nke vertices, na n'ihi ya n-nke ọnụ ọgụgụ nke ndị ọzọ na-akpọ n-gon. Onwe Ya gbajiri akara bụ ókè ma ọ bụ ngwe nke geometric ọgụgụ. Polygonal ụgbọelu ma ọ bụ ewepụghị polygon akpọ akụkụ ikpeazụ nke ihe ọ bụla ụgbọelu, ha na-akpata. N'akụkụ n'akụkụ nke geometric ọgụgụ a na-akpọ polyline agba si otu vertex. Ha ga-abụ ndị agbata obi ma ọ bụrụ na ha na-dabere na iche iche vertices nke polygon.

Ọzọ nkọwa nke convex polygons

Na elementrị jiometrị, e nwere ọtụtụ Ẹkot pụtara nkọwa, na-egosi ihe a na-akpọ a convex polygon. Ọzọkwa, a niile okwu na-dokwara eziokwu. A convex polygon bụ onye nwere:

• ọ bụla nke na-ejikọ abụọ ọ bụla, ihe n'ime ya, ụgha kpamkpam na ya;

• nime ya na-edina ya niile diagonals;

• ọ bụla n'ime ime n'akuku adịghị aka 180 Celsius.

Polygon mgbe niile na-ekewa ụgbọelu abụọ. Otu n'ime ha - na-ejedebeghị (ọ ga-nchọ na a gburugburu), na ndị ọzọ - na-akparaghị ókè. The mbụ a na-akpọ n'ime mpaghara, na nke abụọ - elu nke ebe geometric ọgụgụ. Nke a bụ nrutu nke polygon (ndị ọzọ okwu - ngụkọta akụrụngwa) ọtụtụ ọkara ụgbọ elu. N'ihi ya, onye ọ bụla nke na-enwe nsọtụ na ihe nke dịịrị a polygon kpamkpam bụ nke ya.

Iche convex polygons

Definition convex polygon adịghị egosi na e nwere ọtụtụ di iche iche nke ha. Ma onye ọ bụla n'ime ha nwere ụfọdụ ibiere. N'ihi ya, convex polygons, nke nwere esịtidem n'akuku nke 180 Celsius, kwuru na ubé convex. The convex geometric ọgụgụ nwere atọ ọnụ ọnụ, a na-akpọ a triangle, anọ - quadrilateral, ise - Pentagon, wdg Onye ọ bụla nke convex n-gons-emezu ihe ndị na-esonụ dị mkpa a chọrọ: .. N ga-hà ma ọ bụ ukwuu karịa 3. Onye ọ bụla nke triangles bụ convex. The geometric ọgụgụ nke ụdị nke niile vertices na-emi odude a gburugburu, na-akpọ dere gburugburu. Kọwara convex polygon a na-akpọ ma ọ bụrụ na ya niile n'akụkụ gburugburu a gburugburu metụ ya aka. Abụọ polygons na-akpọ hà naanị na ikpe mgbe na-eji machie nwere ike ikpokọta. Ewepụghị polygon akpọ polygonal ụgbọelu (a ụgbọelu òkè) na nke a mmachi geometrical ọgụgụ.

Ịgachi convex polygons

Ịgachi polygons akpọ geometric shapes na nha anya kwesiri akụkụ na n'akụkụ. Inside ha na e nwere mgbe 0, nke bụ otu anya n'ebe onye ọ bụla nke ya vertices. Ọ na-akpọ n'etiti geometrical ọgụgụ. Lines ijikọ center na vertices nke geometric ọgụgụ a na-akpọ apothem, na ndị na jikọọ ebe 0 na ọzọ - radii.

Correct rektangulu - square. Equilateral triangle a na-akpọ equilateral. N'ihi na ndị dị otú ahụ shapes e ndị na-esonụ na-achị: onye ọ bụla convex polygon n'akuku bụ 180 Celsius * (n-2) / n,

ebe n - ọnụ ọgụgụ nke vertices nke convex geometric ọgụgụ.

The ebe nke ọ bụla mgbe polygon kpebisiri ike site usoro:

S = p * h,

ebe p bụ hà ọkara nchikota nke n'akụkụ niile nke polygon, na h bu ogologo apothem.

Properties convex polygons

Convex polygons nwere ụfọdụ Njirimara. N'ihi ya, nke na-ejikọ abụọ ọ bụla, ihe ndị a geometric ọgụgụ, bụchaghị dị na ya. àmà:

E were ya na P - na convex polygon. Iri abụọ na aka ike ihe, e.g., A na B, nke nwe P. Site ugbu a definition nke a convex polygon, isi ihe ndị a na-emi odude na otu akụkụ nke ogologo akara na e dere ọ bụla direction R. N'ihi ya, AB na-nwere nke a na ihe onwunwe na na-dị na R. A convex polygon mgbe niile nwere ike ekewa n'ime ọtụtụ triangles nnọọ niile na diagonals, nke ẹkenịmde onye nke ya vertices.

Akụkụ convex geometric shapes

The akụkụ nke a convex polygon - bụ akụkụ na-guzobere site ndị ọzọ. Inside nkuku nọ n'ime ebe nke geometric ọgụgụ. The n'akuku nke na-kpụrụ ya n'akụkụ nke converge na a vertex, a na-akpọ n'akuku nke convex polygon. Nkuku n'akụkụ ka esịtidem nkuku nke geometrical ọnụ ọgụgụ, na-akpọ mpụga. Onye ọ bụla n'akụkụ nke a convex polygon, mere ndokwa n'ime ya, bụ:

180 Celsius - x

ebe x - uru n'èzí akuku. Nke a dị mfe usoro bụ na ọdabara ka ụdị ọ bụla nke geometric shapes dị otú ahụ.

Ke ofụri ofụri, maka n'èzí nkuku adị esonụ iwu: ọ bụla convex polygon n'akuku hà dị iche n'etiti 180 Celsius na uru nke ime n'akuku. Ọ nwere ike ụkpụrụ sitere na site -180 ° 180 Celsius. N'ihi ya, mgbe n'ime n'akuku bụ 120 Celsius, ọdịdị ga nwere a uru nke 60 Celsius.

Nchikota nke akụkụ nke convex polygons

Nchikota nke ime akụkụ nke a convex polygon guzosie ike site na usoro:

180 ° * (n-2),

ebe n - ọnụ ọgụgụ nke vertices nke n-gon.

The nchikota nke akụkụ nke a convex polygon na gbakọọ nnọọ nanị. Tụlee ihe ọ bụla dị otú ahụ geometric udi. Iji chọpụta nchikota nke akụkụ na a convex polygon mkpa ijikọ onye nke ya vertices ọzọ vertices. Dị ka a N'ihi ihe a amama (n-2) nke triangle. Ọ maara na nchikota nke akụkụ nke ọ bụla triangle bụ mgbe 180 Celsius. N'ihi na ha na ọnụ ọgụgụ na ihe ọ bụla polygon nhata (n-2), na nchikota nke ime akụkụ nke ọnụ ọgụgụ nhata 180 Celsius x (n-2).

Ikere convex polygon nkuku, ya bụ, ọ bụla abụọ n'akụkụ esịtidem na mpụga akụkụ ha, na nke a convex geometric ọnụ ọgụgụ ha ga-enwe mgbe nile hà 180 Celsius. Dabere na nke a, anyị nwere ike ikpebi na nchikota nke ya niile nkuku:

180 x n.

Nchikota nke ime akụkụ bụ 180 Celsius * (n-2). Ntem, nchikota nke ihe niile elu nkuku nke ọgụgụ setịpụrụ usoro:

180 ° * n-180 Celsius - (n-2) = 360 Celsius.

Nchikota nke mpụga akụkụ nke ọ bụla convex polygon ga-enwe mgbe nile hà 360 Celsius (n'agbanyeghị nke ọnụ ọgụgụ nke ya n'akụkụ).

N'èzí n'akụkụ nke a convex polygon na-adịkarị na-anọchite anya ihe dị iche n'etiti 180 Celsius na uru nke ime n'akuku.

Ndị ọzọ Njirimara nke a convex polygon

E wezụga ihe ndị bụ isi Njirimara nke geometric ọgụgụ data, ha nwere ọzọ, nke-eme mgbe na-ejizi ha. N'ihi ya, ọ bụla nke polygons nwere ike kewaa multiple convex n-gons. Iji mee nke a, na-ọ bụla nke ya n'akụkụ na bee geometric udi tinyere ndị a guzozie e. Kewaa ọ bụla polygon n'ime ọtụtụ convex akụkụ bụ omume na nke mere na n'elu nke ọ bụla n'ime iberibe dabaa n'ụbọchị niile nke ya vertices. Site a geometrical ọgụgụ pụrụ ịbụ nnọọ mfe ka triangles site niile diagonals si otu vertex. N'ihi ya, ọ bụla polygon, n'ikpeazụ, nwere ike kewara a ụfọdụ ọnụ ọgụgụ nke triangles, nke bụ nnọọ uru na idozi dị iche iche ihe aga-eme metụtara ndị dị otú geometrical shapes.

The perimeta nke convex polygon

The agba nke polyline, polygon na-akpọ ndị ọzọ, mgbe mgbe gosiri na ndị na-esonụ letters: ab, BC, cd, de, ea. Nke a n'akụkụ nke a geometrical ọgụgụ na vertices a, b, ch, d, e. Nchikota nke ogologo nke n'akụkụ nke a convex polygon a na-akpọ ya perimeta.

The gbaa nke polygon

Convex polygons nwere ike banye na kọwara. Gburugburu tangent nile n'akụkụ nke geometric ọnụ ọgụgụ, a na-akpọ dere n'ime ya. Nke a polygon a na-akpọ kọwara. The center gburugburu nke e dere na polygon bụ a n'ókè nke nrutu nke bisectors nke akụkụ n'ime a nyere geometric udi. The nke ebe polygon bụ hà:

S = p * r,

ebe r - okirikiri nke e dere gburugburu, na p - semiperimeter a polygon.

A gburugburu nwere polygon vertices, a na-akpọ kọwara nso ya. Ọzọkwa, a na convex geometric ọgụgụ a na-akpọ dere. The gburugburu center, nke a kọwara dị otú ahụ polygon bụ a na-akpọ nrutu ebe midperpendiculars n'akụkụ niile.

Diagonal convex geometric shapes

The diagonals nke a convex polygon - a nke na-ejikọ na-agbata obi vertices. Onye ọ bụla n'ime ha bụ n'ime a geometric ọgụgụ. The ọnụ ọgụgụ nke diagonals nke n-gon atọrọ dị ka usoro:

N = n (n - 3) / 2.

The ọnụ ọgụgụ nke diagonals nke a convex polygon ekere òkè dị mkpa na elementrị jiometrị. The ọnụ ọgụgụ nke triangles (K), nke nwere ike imebi bụla convex polygon, gbakọọ na-esonụ usoro:

K = n - 2.

The ọnụ ọgụgụ nke diagonals nke a convex polygon bụ mgbe dabere na ọnụ ọgụgụ nke vertices.

Nkebi nke a convex polygon

Mgbe ụfọdụ, na-edozi jiometrị aga-eme dị mkpa imebi a convex polygon n'ime ọtụtụ triangles na-abụghị intersecting diagonals. Nsogbu a nwere ike na-edozi site na iwepu a ụfọdụ usoro.

Akọwapụta nsogbu: akpọ ụdị nkebi nke a convex n-gon n'ime ọtụtụ triangles site diagonals na irutu naanị na vertices nke a geometric ọgụgụ.

Ngwọta: e were ya na P1, P2, P3, ..., Pn - n'elu nke n-gon. Number Xn - ọnụ ọgụgụ nke ndị ya partitions. Nlezianya tụlee dapụtara diagonal geometric ọgụgụ Pi Pn. Na ọ bụla nke mgbe partitions P1 Pn bụ nke a akpan akpan triangle P1 Pi Pn, nke 1

Ka i = 2 bụ a otu mgbe partitions, mgbe nile nwere diagonal P2 Pn. The ọnụ ọgụgụ nke partitions na-gụnyere na ya, hà na ọnụ ọgụgụ nke partitions (n-1) -gon P2 P3 P4 ... Pn. Na ọzọ okwu, ọ bụ hà Xn-1.

Ọ bụrụ na i = 3, mgbe ahụ, nke ọzọ otu partitions ga mgbe nile nwere a diagonal P3 P1 na P3 Pn. The ọnụ ọgụgụ nke ziri ezi partitions na-ẹdude ke otu, ga-adaba na ọnụ ọgụgụ nke partitions (n-2) -gon P3, P4 ... Pn. Ndị ọzọ okwu, ọ ga-abụ Xn-2.

Ka i = 4, mgbe ahụ, triangles n'etiti ezi nkebi a na-agbụ na-ebu a triangle P1 Pn P4, nke ga-adjoin na quadrangle P1 P2 P3 P4, (n-3) -gon P5 P4 ... Pn. The ọnụ ọgụgụ nke ziri ezi partitions dị quadrilateral nhata X4, na ọnụ ọgụgụ nke partitions (n-3) -gon nhata Xn-3. Dabere na ndị e kwuru n'elu, anyị nwere ike ikwu na ngụkọta ọnụ ọgụgụ nke mgbe nile partitions na-ẹdude ke otu a nhata Xn-3 X4. Ọzọ dị iche iche, nke i = 4, 5, 6, 7 ... ga-ebu 4 Xn-X5, Xn-5 X6, Xn-6 ... X7 mgbe partitions.

Ka i = n-2, ọnụ ọgụgụ nke ndị ziri ezi partitions na a nyere otu ga-adaba na ọnụ ọgụgụ nke partitions na otu, nke i = 2 (ndị ọzọ okwu, nhata Xn-1).

Ebe ọ bụ na X1 = X2 = 0, X3 = 1 na X4 = 2, ..., ọnụ ọgụgụ nke partitions nke convex polygon bụ:

Xn = Xn-1 + Xn-2 + Xn-3, Xn-X4 + X5 + 4 ... + X 5 + 4 Xn-Xn-X 4 + 3 + 2 Xn-Xn-1.

atụ:

X5 = X4 + X3 + X4 = 5

X6 = X4 + X5 + X4 + X5 = 14

X7 + X5 = X6 + X4 * X4 + X5 + X6 = 42

X7 = X8 + X6 + X4 * X5 + X4 * X5 + X6 + X7 = 132

The ọnụ ọgụgụ nke ziri ezi partitions intersecting n'ime otu diagonal

Mgbe ịlele onye ahụ ikpe, ọ nwere ike na-eche na ọnụ ọgụgụ nke diagonals nke convex n-gon bụ hà ngwaahịa nke niile partitions nke chaatị a ụkpụrụ (n-3).

The gosiri na nke a ọtụtụ ndị chere: e were ya na P1n = Xn * (n-3), mgbe ahụ, ọ bụla n-gon nwere ike kewara (n-2) bụ a triangle. Na nke a otu onye n'ime ha nwere ike stacked (n-3) -chetyrehugolnik. N'otu oge ahụ, onye ọ bụla quadrangle bụ diagonal. Ebe ọ bụ na a convex geometric ọgụgụ abụọ diagonals nwere ike rụrụ, nke pụtara na ihe ọ bụla (n-3) -chetyrehugolnikah nwere ike na-eduzi ọzọ diagonal (n-3). Dabere na nke a, anyị nwere ike ikwubi na mgbe ọ bụla kwesịrị ekwesị nkebi nwere ohere (n-3) -diagonali nzukọ chọrọ nke a ozi.

Area convex polygons

Ọtụtụ mgbe, na idozi dị iche iche na nsogbu nke elementrị jiometrị na e nwere mkpa iji chọpụta ebe ndị a convex polygon. Iche na (nke Iri na Otu. Yi), i = 1,2,3 ... n-anọchi anya a usoro nke na-achịkọta ndị niile ndị agbata obi vertices nke polygon, enweghị onwe-gaferịtara ibe. Na nke a, ya ebe a gbakọọ na-esonụ usoro:

_{S = ½ (Σ (X i} + X i + ₁₎ (Y _{i +} Y _i + _1)),

ebe (X _1, Y _{1) = (X n +1, Y} n + _1).